Modern Physics & Quantum Concepts Cheat Sheet

A printable reference covering photon energy, de Broglie wavelength, photoelectric effect, Bohr model, nuclear decay, and mass-energy equivalence for grades 11-12.

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Modern physics explains phenomena that classical physics cannot fully describe, especially at atomic, nuclear, and high-speed scales. This cheat sheet helps students connect light, matter, energy, and probability using the core formulas of quantum and nuclear physics. It is useful for reviewing photon behavior, electron energy levels, radioactive decay, and relativity before tests or problem sets. The most important ideas include quantized energy, wave-particle duality, and conservation of energy in atomic and nuclear processes. Photon energy is found with $E=hf$ , while matter waves use $\lambda=\frac{h}{p}$ . The photoelectric effect uses $K_{\max}=hf-\phi$ , and nuclear processes often use $E=mc^2$ to connect mass changes with released energy.

Key Facts

Photon energy is given by $E=hf=\frac{hc}{\lambda}$ , where $h$ is Planck's constant, $f$ is frequency, and $\lambda$ is wavelength.
The de Broglie wavelength of a particle is $\lambda=\frac{h}{p}$ , and for nonrelativistic motion $p=mv$ .
In the photoelectric effect, the maximum kinetic energy of emitted electrons is $K_{\max}=hf-\phi$ , where $\phi$ is the work function.
The threshold frequency for photoemission is $f_0=\frac{\phi}{h}$ , so no electrons are emitted when $f<f_0$ .
Bohr energy levels for hydrogen are $E_n=\frac{-13.6\,\text{eV}}{n^2}$ , where $n=1,2,3,\ldots$ .
A photon emitted or absorbed during an atomic transition has energy $\Delta E=hf=\frac{hc}{\lambda}$ .
Radioactive decay follows $N=N_0\left(\frac{1}{2}\right)^{t/T_{1/2}}$ , where $T_{1/2}$ is the half-life.
Mass-energy equivalence is $E=mc^2$ , and nuclear energy released is often calculated from $\Delta E=\Delta mc^2$ .

Vocabulary

Quantum: A quantum is a discrete packet of energy, such as a photon of light with energy $E=hf$ .
Photon: A photon is a particle-like packet of electromagnetic radiation that has energy $E=hf$ and momentum $p=\frac{h}{\lambda}$ .
Work Function: The work function $\phi$ is the minimum energy needed to remove an electron from a material's surface.
de Broglie Wavelength: The de Broglie wavelength is the wavelength associated with a moving particle, given by $\lambda=\frac{h}{p}$ .
Half-Life: Half-life $T_{1/2}$ is the time required for half of the radioactive nuclei in a sample to decay.
Mass Defect: Mass defect $\Delta m$ is the missing mass converted into binding energy in a nuclear system through $\Delta E=\Delta mc^2$ .

Common Mistakes to Avoid

Using intensity instead of frequency to decide if photoelectrons are emitted is wrong because emission requires $hf\ge \phi$ , not just brighter light.
Forgetting to convert electronvolts to joules causes unit errors because $1\,\text{eV}=1.602\times10^{-19}\,\text{J}$ .
Using $\lambda=\frac{h}{mv}$ for photons is wrong because photons have no rest mass, so use $p=\frac{h}{\lambda}$ or $E=pc$ .
Treating Bohr energy levels as positive is incorrect because bound electron energies in hydrogen are negative, with $E_n=\frac{-13.6\,\text{eV}}{n^2}$ .
Subtracting half-lives linearly is wrong because radioactive decay is exponential, so use $N=N_0\left(\frac{1}{2}\right)^{t/T_{1/2}}$ .

Practice Questions

1 A photon has frequency $5.00\times10^{14}\,\text{Hz}$ . Calculate its energy in joules using $E=hf$ .
2 An electron moves at $2.00\times10^6\,\text{m/s}$ . Find its de Broglie wavelength using $\lambda=\frac{h}{mv}$ and $m_e=9.11\times10^{-31}\,\text{kg}$ .
3 A radioactive sample starts with $80.0\,\text{g}$ and has a half-life of $6.0\,\text{days}$ . How much remains after $18.0\,\text{days}$ ?
4 Explain why increasing the brightness of light below the threshold frequency does not cause photoelectrons to be emitted.